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Quantum Cluster State Model with Haagerup Fusion Category Symmetry

Zhian Jia

Abstract

We propose a (1+1)D lattice model, inspired by a weak Hopf algebra generalization of the cluster state model, which realizes Haagerup fusion category symmetry and features a tensor product Hilbert space. The construction begins with a reconstruction of the Haagerup weak Hopf algebra $H_3$ from the Haagerup fusion category, ensuring that the representation category of $H_3$ is equivalent to Haagerup fusion category. Utilizing the framework of symmetry topological field theory (SymTFT), we develop an ultra-thin weak Hopf quantum double model, characterized by a smooth topological boundary condition. We show that this model supports Haagerup fusion category symmetry. Finally, we solve the ground state of the model in terms of a weak Hopf matrix product state, which serves as a natural generalization of the cluster state, embodying Haagerup fusion category symmetry.

Quantum Cluster State Model with Haagerup Fusion Category Symmetry

Abstract

We propose a (1+1)D lattice model, inspired by a weak Hopf algebra generalization of the cluster state model, which realizes Haagerup fusion category symmetry and features a tensor product Hilbert space. The construction begins with a reconstruction of the Haagerup weak Hopf algebra from the Haagerup fusion category, ensuring that the representation category of is equivalent to Haagerup fusion category. Utilizing the framework of symmetry topological field theory (SymTFT), we develop an ultra-thin weak Hopf quantum double model, characterized by a smooth topological boundary condition. We show that this model supports Haagerup fusion category symmetry. Finally, we solve the ground state of the model in terms of a weak Hopf matrix product state, which serves as a natural generalization of the cluster state, embodying Haagerup fusion category symmetry.
Paper Structure (1 section, 1 theorem, 53 equations, 1 figure, 3 tables)

This paper contains 1 section, 1 theorem, 53 equations, 1 figure, 3 tables.

Table of Contents

  1. Details of Haagerup boundary tube algebra $H_3$

Key Result

Theorem 1

By applying the Tannaka-Krein reconstruction or the boundary tube algebra approach, we can recover a $C^*$ weak Hopf algebra $H_3$, whose representation category $\mathsf{Rep}(H_3)$ is equivalent to the Haagerup fusion category $\mathcal{H}_3$ as fusion categories. When $H_3$ is used as input data f Note that $\mathcal{H}_3 \simeq \mathsf{Rep}(H_3) \subset \operatorname{Cocom}(\hat{H}_3) \subset \

Figures (1)

  • Figure 1: Illustration of SymTFT sandwich and cluster ladder model. (a) The symmetry TFT consists of a symmetry boundary $\EuScript{B}_{\rm sym}$ which encodes the fusion category symmetry; a physical boundary, which may be gapped or gapless that encodes the dynamics of the theory; the bulk is a topological field theory $\text{Z}(\EuScript{B}_{\rm sym})$. (b) Depiction of the cluster ladder model, which is an ultra-thin quantum double model with two boundaries (the qudit is put on edges), one boundary is chosen as a smooth boundary that encodes the symmetry information. If the physical boundary is chosen as a rough boundary, then the model becomes a cluster state model. (c) The chessboard representation of the cluster ladder model, where the qudits are placed on vertices, and each vertex corresponds to an edge of the quantum double model. For the cluster state model, the vertices on the physical boundary (cyan vertices) are removed, leaving only two types of local stabilizers: one for the symmetry boundary vertex operator and the other for the face operator.

Theorems & Definitions (1)

  • Theorem 1